Chapter 11.3, Problem 62STP

To determine

To find the factors of:

  $x^3-4x$

Answer to Problem 62STP

Option D.

Explanation of Solution

Given:

The options are:

  $\begin{array}{l}\text{A}\text{. }{x}^3,-4x\text{ B}\text{. }x,x^2-4\\ \text{C}\text{. }x-2,x+2\text{ D}\text{. }x,x-2,x+2\end{array}$

Concept Used:

To factor a polynomial completely:

1. Factor out the greatest monomial factor first.
2. Look for a difference of squares and a perfect square trinomial.
3. If a trinomial is not a square, look for a pair of binomial factors.
4. Make sure that each binomial or trinomial factor is prime.

Calculation:

In order to find the factors of:

  $x^3-4x$

First take the common factor of each term x out, as

  $\begin{array}{c}{x}^3-4x=x\cdot x^2-4\cdot x\\ =x\left(x^2-4\right)\\ =x\left(x^2-2^2\right)\end{array}$

Now, factorize the term $\left(x^2-2^2\right)$ using difference of square formula $\left(a^2-b^2\right)=\left(a-b\right)\left(a+b\right)$ .

As,

  $\left(x^2-2^2\right)=\left(x-2\right)\left(x+2\right)$

Thus, the factors of polynomial $x^3-4x$ are $x,\left(x-2\right),\text{ and }\left(x+2\right)$ .

Thus, option D is correct.